Elementary Algebra Chapter 3

Linear Equations and Inequalities in Two Variables

 

Some good supplemental material as well as worked examples are available HERE

From Section 3.1 Graphing Using the Rectangular Coordinate System

Definition: A rectangular coordinate system (Cartesian coordinate system) consists of two perpendicular number lines. One number line is drawn horizontally and the other is drawn vertically.

Definition: The horizontal number line is usually called the x-axis.

Definition: The vertical number line is usually called the y-axis.

Definition: The point of intersection of the two number lines is called the origin of the coordinate system.

Definition: The two axes form a coordinate plane and divide it into four quadrants named Quadrant I, Quadrant II, Quadrant III, and Quadrant IV as shown in the diagram here.

Definition: Each point in a coordinate plane can be identified by a pair of real numbers x and y written in the form (x, y). The first number is called the x-coordinate and represents a number on the horizontal number line. The second number is called the y-coordinate and represents a number on the vertical number line. The numbers in the pair are called the coordinate of the point.

 

Definition: The pair (x, y) is called an ordered pair because the order of the two numbers inside the parenthesis makes a difference.

 

Example: The ordered pair (3, 4) represents a different point than the ordered pair (4, 3)

Definition: The process of locating a point in the coordinate plane is called graphing or plotting the point.

 

Procedure: To plot a point (a, b):

Construct a vertical line through the number a on the x-axis.

Construct a horizontal line through the number b on the y-axis.

The point where these two lines intersect is the graph of the point (a, b).

 

Procedure: To determine the coordinate of a point in the coordinate plane:

Construct a vertical line through the point. It will intersect the x-axis at some number k.

This number k is the x-coordinate of the point.

Construct a horizontal line through the point. It will intersect the y-axis at some number t.

The number t is the y-coordinate of the point.

The coordinate of the point are (k, t).

From Section 3.2 Graphing Linear Equations

Definition: A solution of an equation in two variables x and y is an ordered pair of real numbers (d, f) whose coordinates make the equation a true statement when the first coordinate is substituted for x and the second coordinate is substituted for y in the equation. We say the point (d, f) satisfies the equation.

 

Definition: The graph of an equation consists of all the points, and only those points, which are solutions of the equation.

An alternately, but equivalent definition is: The graph of an equation consists of all the points, and only those points, which satisfy the equation.

 

From Section 3.3 More About Graphing Linear Equations

Click here for a complete discussion of Linear Equations

Definition: A linear equation is an equation which may be written in the form y = mx + b where m, and b are real numbers.

Definition: The General Form for the equation of a line is Ax + By = C where A, B, and C are real numbers and not both A and B are zero.

Definition: The y-intercept of a line is the point (0, b) where the line intersects the y-axis.

Definition: The x-intercept of a line is the point (a, 0) where the line intersects the x-axis.

 

Remark: The equation y = b represents a horizontal line with y-intercept b.

Remark: The equation x = a represents a vertical line with x-intercept a.

 

Procedure: To graph a linear equation:

Find two pairs (x, y) that satisfy the equation.

Plot the two pairs

Draw a line through the two plotted points.

 

From Section 3.4 The Slope of a Line

Definition: A ratio is the quotient of two numbers.

Definition: Ratios that are used to compare quantities with different units are called rates.

Definition: The slope of the non-vertical line through two points (x1, y1) and (x2, y2) is

Remark: Slope is undefined for vertical lines.

Remark: Slope is 0 for horizontal lines

 

From Section 3.5 The Slope-Intercept Form of the Equation of a Line

Definition: Slope-Intercept Form of the Equation of a Line

If a linear equation is written in the form y = mx + b where m and b are real numbers, the graph of the equation is a line with slope m and y-intercept (0,b)

Definition: Two lines are parallel if they have the same slopes and different y-intercepts.

Definition: Two lines are perpendicular if their slopes are negative reciprocals of each other and conversely if the slopes of two lines are negative reciprocals, the lines are perpendicular.

Remark: The fact that two slopes m1 and m2 are negative reciprocals of each other may be expressed algebraically in the each of the following ways.

From Section 3.6 The Point-Slope Form of the Equation of a Line

Definition: Point-Slope Form of the Equation of a Line

If a line with slope m passes through the point (x1, y1) the equation of the line is y y1 = m(x x1).

 

 

 

From Section 3.7 Graphing Linear Inequalities

Definition:A linear inequality in two variables x and y is an inequality which can be written as
Ax + By < C, Ax + By > C, Ax + By = C, or Ax + By = C.

Definition: A point (x, y) is a solution of an inequality in two variables if the coordinates satisfy the inequality. (that is, if a true statement results when the coordinates are substituted for the variables in the inequality.)

Definition: The collection of all solutions of an inequality is called the solution set of the inequality.

Definition: The graph of an inequality is the set of points which are solutions of the inequality. (That is, the graph is the set of all points whose coordinates satisfy the inequality).

Definition: If the inequality symbol in an inequality in two variables is replaced with an equality symbol, the graph of the resulting equation is called the boundary line for the inequality.

Fact: The graph of an inequality in two variables is a half-plane.

Fact: The boundary line forms the boundary between the half-plane consisting of all solutions of the inequality and the half-plane consisting of all points which are not solutions of the inequality

Procedure: To graph a linear inequality in two variables:
a) Sketch the graph of the boundary line
      i) as a solid line if the inequality symbol is either or .
     ii) as a dashed line if the inequality symbol is either > or <.
b) Pick a point, not on the boundary line, as a test point and substitute its coordinates into the inequality.
c) If the result from Step b is a TRUE statement, the half-plane containing the test point is the solution set.
d) If the result from Step b is a FALSE statement, the half-plane which does not contain the test point is the solution set.
e) Shade the half-plane which is the solution and label all important points.